---
date: 2021-04-29
tags: [ web/quick-notes ]
---

# Ribet, "Class groups and Galois representations"

Tags: #projects/notes/seminars 

> Reference: Ribet, "Class Groups and Galois Representations". 
> <https://math.berkeley.edu/~ribet/herbrand.pdf>

- Alternate definition of [Unsorted/class group](Unsorted/class%20group.md) : the group of fractional ideals.
 	- Defined as $\ZZ[\mspec \OO_K]$ (the free $\ZZ\dash$module on maximal ideals) modulo principal [fractional ideal](fractional%20ideal.md)

- What *is* the [maximal unramified extension](maximal%20unramified%20extension.md), i.e. the [Hilbert class field](Hilbert%20class%20field.md)?

- The [Artin map](Artin%20map.md) from [class field theory](class%20field%20theory.md) : $\Cl(K) \mapsvia{\sim} \Gal(H/K)$ where $\mfp \mapsto \Frob_{\mfp}$.

- Set $G_k \da \Gal(\Qbar/K)$, then $\Cl(K)$ is a quotient of $G_k^{\Ab}$.

  - Equivalently, $\Cl(K)\dual \leq G_k\dual$ where $(\wait)\dual \da \Hom_{\Top\Grp}(\wait, \CC\units)$.
  	I.e. take continuous [characters](characters.md).

- Open question: are there infinitely many [quadratic fields](quadratic%20fields) $K$ for which $\Cl(K) = 0$

- [Dedekind zeta function](Dedekind%20zeta%20function.md)

\[
\zeta_K \da \prod_{\mfp \in \mspec \OO_K}(1 - N(\mfp)^{-s} )\inv
.\]

  - Note: guessing about the indexing set here.
  Original source just indexes over $\mfp$...

  - $\Res_{s=1} \zeta_K$ involves $h_k \da \size \Cl(K)$.

- Serre stresses: use functional equation to look at $s=0$ instead of $s=1$!
  Leads to cleaner/simpler formulas.

- [Kummer theory](Kummer%20theory.md) proved FLT for exponent $p$ for **regular** primes, i.e. $\gcd(h_K, p) = 1$.
  - Kummer's criterion: $p$ is regular iff $p$ divides none of the numerators of some [Bernoulli numbers](Bernoulli%20numbers).

- What is the [Teichmuller character](Teichmuller%20character)?

- See [Birch and Swinnerton-Dyer conjecture](Birch%20and%20Swinnerton-Dyer%20conjecture.md)
	- Define the [Tate-Shafarevich group](Tate-Shafarevich%20group.md) as $\Sha(E/\QQ)$ for an [elliptic curve](elliptic%20curve.md), then

\[
\size \Sha(E/\QQ) \underset{?}{=} \qty{ L(E, 1) \over \Omega} \qty{ \size(E(\QQ))^2 \over \prod_\ell w_\ell}
,\]

  where \( \Omega \) is a [period](period.md) and \( w_\ell \) are the local [Tamagawa numbers](Tamagawa%20numbers.md).

- Need lower bounds of sizes of [class groups](class%20group.md).
  Might be able to use elliptic curves, congruences between [Eisenstein series](Eisenstein%20series.md) and [cusp forms](cusp%20forms) on $\operatorname{U}(2, 2)$ -- can obtain 4-dimensional [Galois representations](Galois%20representations.md) that lead to nontrivial elements of $\Sha$.